Question

calcule essa integral

Answer 1

$\int\limits^\infty_1 {\frac{1}{x^3+x} } \, dx$

$\lim_{a \to +\infty} (\int\limits^a_1 {\frac{1}{x^3+x} } \, dx )$

$\lim_{a \to +\infty}( \int\limits {\frac{1}{x}-\frac{x}{x^2+1} } \, dx )$

$\lim_{a \to +\infty}( \int\limits {\frac{1}{x} } \, dx-\int\limits {\frac{x}{x^2+1} } \, dx )$

$\lim_{a \to +\infty}( ln( |x | )-\frac{1}{2}\times ln(x^2+1) )$

$\lim_{a \to +\infty}( ln( |a | )-\frac{1}{2}\times ln(a^2+1)-(ln(|1|)-\frac{1}{2}\times ln(x^2+1)) )$

$\lim_{a \to +\infty}( ln( |a | )-\frac{1}{2}\times ln(a^2+1)+\frac{1}{2}\times ln(2) )$

$\lim_{a \to +\infty}( ln( a )-\frac{1}{2}\times ln(a^2+1)+\frac{1}{2}\times ln(2) )$

$\lim_{a \to +\infty}( ln( a )-\frac{1}{2}\times ln(a^2+1))+ \lim_{a \to +\infty} (\frac{1}{2}\times ln(2) )$

$0+ \frac{1}{2}\times ln (2)$

$\frac{1}{2} \times ln(2) ~ou~ \approx 0,346574$